Question 252431
the 2 equations are in standard form of ax + by = c


the slope intercept form of these equations would be y = mx + b where m is the slope and b is the y-intercept (value of y when x = 0).


the easiest thing for you to do is to transform the standard form of these equations into the slope-intercept form of them.


all you do is solve for y and this happens automatically.


your first equation is:


x + 2y = 3


subtract x from both sides of this equation to get:


2y = -x + 3


divide both sides of this equation by 2 to get:


y = -(1/2)*x + (3/2)


your slope is -(1/2).
your y-intercept is (3/2).


your second equation is:


3y + Ax = 2


subtract Ax from both sides of this equation to get:


3y = -Ax + 2


divide both sides of this equation by 3 to get:


y = -(A/3)*x + (2/3)


your slope is -(A/3).
your y-intercept is (2/3).


in order for the lines formed by these equations to be perpendicular, the slopes have to be negative reciprocals of each other.


the two slopes you have to work with are:


-(1/2) and -(A/3).


the negative reciprocal of a number is equal to -1 divided by the number.


the negative reciprocal of -(1/2) = -1/-(1/2).


this comes out to be equal to 2.


in order for the lines of these equations to be perpendicular to each other, -(A/3) must be equal to 2 which is the negative reciprocal of -(1/2).


your equation to solve is:


-(A/3) = 2


multiply both sides of this equation by 3 to get:


-A = 6


multiply both sides of this equation by (-1) to get:


A = -6


your answer is A = -6.


your original equations were:


x + 2y = 3 and 3y + Ax = 2 


replace A with -6 to get:


x + 2y = 3 and 3y - 6x = 2


to graph these equations, we need to solve for y which automatically puts them into the slope-intercept form.


we get:


y = -(1/2)*x + (3/2) and y = (6/3)*x + (2/3)


the second equation simplifies to:


y = 2*x + (2/3) because 6/3 is the same as 2.


the graph of these equations looks like this:


{{{graph (600,600,-10,10,-10,10,2*x + (2/3),-(1/2)*x + (3/2))}}}


y = -(1/2)*x + (3/2) crosses the y-axis at (3/2) = 1.5.   This graph slopes down from left to right.


y = 2*x + (2/3) crosses the y-axis at (2/3) = .66666667.   This graph slopes up from left to right.